Ken, is an "indestructable" number only one that is fixed under the powertrain mapping, or is a number "indesctructible under mapping M(f)" if M(x) = x? If so, does the term apply to any generic set element under a mapping of element type to the same element type?

Ken, is an "indestructable" number only one that is fixed under the powertrain mapping, or is a number "indesctructible under mapping M(f)" if M(x) = x? If so, does the term apply to any generic set element under a mapping of element type to the same element type?

So the first answer is: I don't know. I first heard of indestructible numbers by reading the "interesting numbers" thread and following up by googling Below and Conway on it all. So I don't know nothin.

But I am not sure that I understand your question. Yes I am (reasonably) sure that the term is context specific, meaning that Conway or Bello or one of those guys invented it to deal with numbers arising in some manner from the powertrain map. I say this just on a general feel for how terms like this come into being. Sometimes it seems there is a gnome locked away somewhere making up these terms.

Someone recently sent me a video related to this perverse invention of words:

what happens to the ships traveling the Pacific when the tsunamis come to them?

As I understand it, hardly noticeable in deep water--a gentle one-meter rise in water level. It's when the tsunami reaches shallow areas that the huge wall of water forms.

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As I understand it, hardly noticeable in deep water--a gentle one-meter rise in water level. It's when the tsunami reaches shallow areas that the huge wall of water forms.

Exactly. Tsunami (japanese for "harbor wave") waves have a large wave length (tens of kilometers), so that they are a very large and shallow bulge. In deep water, this is not much of a problem and completely harmless to a ship.

However, the speed of the wave decreases with depth in shallower water (you can notice this at the beach when waves from the deep water come in at an angle, the part closest to the shore will show down so it appears the wave turns its wake parallel to the coast). This means that the back of the wave starts to catch up with the front, leading to piling up of water and to the wall of water that we know as a tsunami.

As the wave displaces such a large volume of water, before the top reaches the shore the low point of the wave reaches it first and can be observed as a sudden quick low tide.

sr but I don't see the purpose of this thread. If you have a question isn't it more practical to post it as a separate topic? Then people can decide whether to read it or not depending on the topic title. And we don't get confusion with discussions about one topic mixed up with other topics.

For the same reason I dislike threads of the type "a couple of hands from yesterday". Just post each hand as separate topic (unless they have a common theme, e.g. both were about whether to make an offshape 1NT opening, or both were about whether to cover an honor lead from dummy).

... most of the new ideas I get are pretty "boring", mostly focusing on constructive methods rather than destructive ones --- Kungsgeten

sr but I don't see the purpose of this thread. If you have a question isn't it more practical to post it as a separate topic? Then people can decide whether to read it or not depending on the topic title. And we don't get confusion with discussions about one topic mixed up with other topics.

For the same reason I dislike threads of the type "a couple of hands from yesterday". Just post each hand as separate topic (unless they have a common theme, e.g. both were about whether to make an offshape 1NT opening, or both were about whether to cover an honor lead from dummy).

If you want you can still post it as a separate topic, if you feel like it's maybe not such an important question, or for whatever other reasons you may have, you can post it here.

Yes. There are various questions that occur to me that don't really seem to warrant the dignity of having their own dedicated thread but still I might like to ask. "What's the name of that paper and pencil game with the boxes?" seems right. Anyway, I'll keep checking in.

ad infinitum. the powers of all other last digits have the same period of 4 behaviour (in addition, 0, 1, 5 and 6 remain constant; 4 and 9 have a period of 2; 2, 3, 7, 8 are the ones where the smallest period is 4). Is there some fancy explanation to this or is this just a combination of a coincidence and its consequences?

ad infinitum. the powers of all other last digits have the same period of 4 behaviour (in addition, 0, 1, 5 and 6 remain constant; 4 and 9 have a period of 2; 2, 3, 7, 8 are the ones where the smallest period is 4). Is there some fancy explanation to this or is this just a combination of a coincidence and its consequences?

You already know that the last digit is dependant only in the previous last digit and the base of the power, so you already know the answer, why do you ask?

ad infinitum. the powers of all other last digits have the same period of 4 behaviour (in addition, 0, 1, 5 and 6 remain constant; 4 and 9 have a period of 2; 2, 3, 7, 8 are the ones where the smallest period is 4). Is there some fancy explanation to this or is this just a combination of a coincidence and its consequences?

Yes, there is a fancy explanation (both link to wikipedia). Given any base "b" you are correct in realizing that all that matters is the last digit. Another way of saying this is that what matters is the remainder of the number when you divide by "b".

One can do arithmetic on the numbers from 0 to b-1 (the remainders). It's referred to as "clock arithmetic" sometimes or "modular arithmetic." If you add to numbers, you imagine "wrapping around." For example, if "b = 5" then "2 + 4 = 6 = 1" (since the remainder of 6 and 1 is the same when divided by 5). We can also multiply "3*4 = 12 = 2" (since 12 and 2 have the same remainder when divided by 5).

You may see what this has to do with your problem. We want to know the first time that 2*2*2....*2 = 2 (with b = 10). There is a theorem that if "b" is prime then this number must always divide b-1. In general it will divide the Euler Totient function of "b." To define this we assume that "b" factors into products of powers of prime numbers as:

b = p_1^(e_1)*p_2^(e_2)...*p_k^(e_k)

Then the Euler Totient function of "b" is

(p_1-1)*...*(p_k-1)*p_1^(e_1-1)*p_2^(e_2-1)...*p_k^(e_k-1)

As an example with b = 10. It factors as 2*5, so the totient function is (2-1)*(5-1) = 4. This is why all the "periods" you see divide 4.

I fear I may have given too much detail (and at once not enough), but I'm a bit too tired to edit this so please ask follow up questions if you want more details about parts of this.

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cool, I will need to read a little more on this, but this has been bugging me on and off for the last 8 years or so! I never could find a good way of googling it and when my mathematician roommate explained to me, I didn't understand anything and I didn't want to ask twice (out of shame or laziness).

So the first answer is: I don't know. I first heard of indestructible numbers by reading the "interesting numbers" thread and following up by googling Below and Conway on it all. So I don't know nothin.

But I am not sure that I understand your question. Yes I am (reasonably) sure that the term is context specific, meaning that Conway or Bello or one of those guys invented it to deal with numbers arising in some manner from the powertrain map. I say this just on a general feel for how terms like this come into being. Sometimes it seems there is a gnome locked away somewhere making up these terms.

I was just wondering if the term was generic (having a name for an element that is invariant when a particular mapping is applied seems useful, for low-grade versions of useful). And if it was generic, it needn't apply solely to numbers; a function could be indestructible when a mapping M(function)->function is applied, and the same could be of any element of any set and its mappings.

And this is why I'm an engineer and not a pure math guy - because this is interesting, but not useful, to me.